Solve for $x$ : $ 7|x - 2| - 10 = -3|x - 2| + 8 $
Answer: Add $ {3|x - 2|} $ to both sides: $ \begin{eqnarray} 7|x - 2| - 10 &=& -3|x - 2| + 8 \\ \\ { + 3|x - 2|} && { + 3|x - 2|} \\ \\ 10|x - 2| - 10 &=& 8 \end{eqnarray} $ Add ${10}$ to both sides: $ \begin{eqnarray} 10|x - 2| - 10 &=& 8 \\ \\ { + 10} &=& { + 10} \\ \\ 10|x - 2| &=& 18 \end{eqnarray} $ Divide both sides by ${10}$ $ \dfrac{10|x - 2|} {{10}} = \dfrac{18} {{10}} $ Simplify: $ |x - 2| = \dfrac{9}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 2 = -\dfrac{9}{5} $ or $ x - 2 = \dfrac{9}{5} $ Solve for the solution where $x - 2$ is negative: $ x - 2 = -\dfrac{9}{5} $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& -\dfrac{9}{5} \\ \\ {+ 2} && {+ 2} \\ \\ x &=& -\dfrac{9}{5} + 2 \end{eqnarray} $ Change the ${ + 2}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{9}{5} {+ \dfrac{10}{5}} $ $ x = \dfrac{1}{5} $ Then calculate the solution where $x - 2$ is positive: $ x - 2 = \dfrac{9}{5} $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& \dfrac{9}{5} \\ \\ {+ 2} && {+ 2} \\ \\ x &=& \dfrac{9}{5} + 2 \end{eqnarray} $ Change the ${ + 2}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{9}{5} {+ \dfrac{10}{5}} $ $ x = \dfrac{19}{5} $ Thus, the correct answer is $x = \dfrac{1}{5} $ or $x = \dfrac{19}{5} $.